[Math] How to find all primitive triples (a,b,c)? (Pythagorean Triples)

elementary-number-theorynumber theorypythagorean triples

how to find all primitive triples when one value of $(a,b,c)$ is given? For example in this case $a = 45$.

What is the procedure to find the primitive triples ?

Conditions for primitive triples are:

  1. $a^2 + b^2 = c^2$

  2. $(a,b) = 1$

  3. $a$ and $c$ are odd and $b$ is even.

Best Answer

There is a well-known parametrization of primitive triples as $(n^2-m^2, 2mn, n^2+m^2)$ for appropriate values of $n,m$. So for your problem you should proceed by solving for $n$ and $m$ coprime such that $n^2-m^2 = (n+m)(n-m) = 45$.

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